Conventionally, there have been known mechanical resonators employing micro mechanical vibrators (micro mechanical resonators or MEMS resonators) (Patent Literature 1).
FIG. 17A and FIG. 17B are views illustrating an example of the structure of a conventional MEMS resonator 100. The MEMS resonator 100 is a so-called capacitance-type MEMS resonator. FIG. 17A is a perspective view of the MEMS resonator 100, and FIG. 17B is a side cross-sectional view of the MEMS resonator 100 taken along the line A-A′ in FIG. 17A. Further, in FIG. 17B, there are not illustrated a BOX (Buried Oxide) layer 104 and a silicon substrate 105, while there are additionally illustrated a voltage Vi inputted to the MEMS resonator 100, an electric current io outputted therefrom, a bias voltage Vp applied to a vibrator 101, the direction of vibrations of the vibrator 101, and the like.
The MEMS resonator 100 can be fabricated using an SOI (Silicon On Insulator) substrate. In this case, the beam-type vibrator 101, an input electrode 102 and an output electrode 103 are formed from the uppermost Si layer in the SOI substrate. Further, the BOX (Buried Oxide) layer 104 under the vibrator 101 has been etched away, and the vibrator 101 is held by a supporting portion 101s on a remaining portion of the BOX layer 104, such that the vibrator 101 can vibrate. The vibrator 101 is anchored, together with the electrodes 102 and 103, to the silicon substrate 105, through the remaining portion of the BOX layer 104.
The mechanism for vibrating the vibrator 101 will be described, with reference to FIG. 17B. The vibrator 101 is placed such that it faces the input electrode 102 and the output electrode 103 with cavities (gaps) gi and go interposed therebetween, and the bias voltage Vp is applied to the vibrator 101 in such a way as to provide a DC electric-potential difference between the input electrode 102 and the output electrode 103. When the AC input voltage (AC voltage) Vi is applied to the input electrode 102, the electric-potential difference between the vibrator 101 and the input electrode 102 is changed according to the AC input voltage Vi, which exerts, on the vibrator 101, an excitation force caused by an electrostatic force. When the frequency of the AC input voltage Vi is coincident with the mechanical resonance frequency of the vibrator 101, the vibrator 101 largely vibrates (resonates), in the direction of vibrations 106. At this time, a displacement current io is flowed to the output electrode 103 from the capacitance Co formed by the cavity go.
Applications of the MEMS resonator 100 include filter circuits which utilize the fact that electricity passing characteristics between the input and output electrodes are increased only around a certain frequency, namely around the resonance frequency of the vibrator, temperature sensors which utilize the fact that the resonance frequency of the vibrator is shifted with the temperature, pressure sensors which utilize the fact that the resonance frequency of the vibrator is shifted due to stresses acting on the vibrator, mass sensors which utilize the fact that the resonance frequency of the vibrator is shifted due to minute quantities of substances adhered to the vibrator, and the like.
Non-Patent Literature 1 suggests a possibility of realization of a pressure sensor employing a MEMS resonator. According to the same, the vibrating motion (for example, resonating motion) of the vibrator in the MEMS resonator is changed, in terms of its characteristics (for example, the Q factor and the magnitude of the amplitude of the vibrating motion), depending on the pressure of the ambient atmosphere surrounding the vibrator. More specifically, in the MEMS resonator, the kinetic energy or the kinetic momentum of the vibrator performing resonant motion is dissipated through the viscosity of the ambient atmosphere surrounding the vibrator, and the degree of the dissipation is varied depending on the pressure of the ambient atmosphere. Therefore, the amplitude of the vibrator resonating at the resonance frequency is varied depending on the pressure of the ambient atmosphere. Accordingly, in the MEMS resonator resonating around the resonance frequency, the amplitude of the vibrator, the Q factor and other quantities thereof well correspond to the pressure of the ambient atmosphere. Accordingly, by detecting the amplitude of the vibrator or the Q factor thereof, in the MEMS resonator resonating in the ambient atmosphere, it is possible to determine the pressure of the ambient atmosphere. For example, FIG. 4 in Non-Patent Literature 1 illustrates a correspondence relationship between the Q factor of the MEMS resonator and the pressure of the ambient atmosphere.
Further, Non-Patent Literature 2 describes nonlinear behaviors of a MEMS resonator which appear in cases where the vibrator in the MEMS resonator is vibrated at relatively-larger amplitudes. In general, when the vibration amplitude of the vibrator 101 in the MEMS resonator 100 is sufficiently smaller, such nonlinear effects have small influences enough to be negligible (in a linear region), and its resonance characteristic obtained by sweeping of the frequency of the input voltage Vi has a profile which is bilaterally symmetric about a peak at the resonance frequency f0 of the vibrator 101 and, thus, exhibits no hysteresis depending on the direction of the sweeping, as a resonance characteristic 111 in FIG. 18. However, if the Q factor of the vibrator is increased so that its vibration amplitude is increased to be equal to or more than a certain magnitude (if it enters a nonlinear region), its resonance characteristic (such as resonance curves 121 and 131) exhibits prominent nonlinearity, as illustrated in FIG. 19 and FIG. 20. For example, when the vibrator 101 in the capacitance-type MEMS resonator 100 is performing vibrating motion in the nonlinear region, its resonance characteristic exhibits hysteresis (123 and 125) depending on the direction of the frequency sweeping, and its vibration amplitude has no obvious peak at the resonance frequency f0.
According to Non-Patent Literature 2, such nonlinear phenomena are caused by two types of nonlinear effects. One of them is an effect (Capacitive Bifurcation) of causing the input electrode 102 and the output electrode 103 to excessively draw the vibrator 101 thereinto, when the vibration amplitude of the vibrator 101 is larger. The other one of them is an effect (Mechanical Bifurcation) of the increase of the rigidity of the vibrator 101 along with the increase of the vibration amplitude of the vibrator 101. Only one of these two types of nonlinear effects may be induced, and, also, both of them may be induced at the same time, depending on the structure of the MEMS resonator.
FIG. 19 is an example of a resonance characteristic 121 of the MEMS resonator 100, when there is prominent capacitive bifurcation. In this case, the resonance characteristic is warped such that it falls leftwardly (toward a lower-frequency side) and, thus, exhibits hysteresis (arrows 123 and 125) due to the difference of the direction of the frequency sweeping, and the peak of the vibration amplitude is shifted toward a lower frequency than the resonance frequency f0.
FIG. 20 is an example of a resonance characteristic 131 of the MEMS resonator, when there is prominent mechanical bifurcation. In this case, the resonance characteristic is warped such that it falls rightwardly (toward a higher-frequency side) and, thus, exhibits hysterisis (arrows 133 and 135) due to the difference of the direction of the frequency sweeping, and the peak of the vibration amplitude is shifted toward a higher frequency than the resonance frequency f0.